Fourier transforms are used widely in chemistry and allied sciences. Examples include infrared, nuclear magnetic resonance, and mass spectroscopies. A thorough understanding of Fourier methods assists the understanding of microscopy, X-ray diffraction, and diffraction gratings. The theory of Fourier transforms has been presented in this "Journal", but detailed practical exercises are limited. A lab-based experiment is described that begins with the acquisition of an interferogram, provides background and instructions for computing and interpreting transformed data using widely available software, applies the principles of Connes' advantage and sampling theory to calibrate the spectrum, and finishes with an infrared spectrum. Related treatment of free induction decay data from an NMR spectrometer is provided in Supporting Information. (Contains 1 table and 6 figures.)